pitmonticone / unitrento_summerschool_geometricanalysis_2023

Repository hosting resources for the summer school "Three Days between Analysis and Geometry" at the University of Trento (@UniTrento) from 2023/08/28 to 2023/08/30.

Summer School in Geometric Analysis (Trento, 2023)

• What: "Three Days between Analysis and Geometry in Trento"
• When: 2023/08/28 - 2023/08/30
• Where: University of Trento, Italy
• Who: Organised by the Department of Mathematics, University of Trento

Lectures

Elia Bruè (Bocconi University)

Title

The fundamental groups of manifolds with nonnegative Ricci curvature.

Abstract

In 1968, Milnor proposed a conjecture stating that Riemannian manifolds with nonnegative Ricci curvature have finitely generated fundamental groups. The first part of the course will focus on comprehending this conjecture and discussing the progress made in addressing it throughout the years. In the subsequent part, we will discuss recent work with Naber and Semola, where we provide a counterexample to Milnor's conjecture. More specifically, we construct a seven-dimensional manifold with nonnegative Ricci curvature whose fundamental group is isomorphic to $\mathbb{Q}/\mathbb{Z}$. The key ideas behind the construction will be explained, and the resulting manifold's geometry will be described in detail.

Resources

• Paper (Website, PDF)

Adam Parusinski (Côte d'Azur University)

Title

Introduction to semialgebraic and subanalytic geometry.

Abstract

The goal of this course is to give an introduction to semialgebraic and subanalytic sets and mappings with emphasis on the properties used in various areas of differential geometry and analysis. In particular we cover such subjects as Łojasiewicz Inequalities, decompositions and stratifications. One lecture will be devoted to the metric properties of these sets including the L-regular decomposition, subanalytic preparation theorem and Lipschitz stratification. We also discuss the integration on subanalytic sets, local densities, and the dependence of integrals on parameters in families.

Resources

• Lecture notes (Website, PDF)

Talks

Luigi De Rosa (University of Basel)

Title

Intermittency and Minkowski content in turbulence.

Abstract

In 1941 Kolmogorov theorized that all p-th moments of increments of the velocity in a turbulent flow have 1/3 has a universal regularity exponent. However, downward deviations from K41 prediction are experimentally observed. This phenomenon is nowadays known as "intermittency", which theoretical physicists, starting from Landau, linked to the spottiness of the region where the dissipation is supported. We propose a couple of Minkowski-type notions of dimensions, one Eulerian and one Lagrangian, which lay down a setup to make Landau's objection quantitative. The approach is quite geometrical and it is in fact part of a more general picture in which most of the PDEs in fluid dynamics fall.

Resources

• Paper (Website, PDF)

Stefano Spirito (University of L'Aquila)

Title

On the inviscid limit for 2D incompressible fluid.

Abstract

We review some recent results concerning the inviscid limit for the 2D Euler equations with irregular vorticity. In particular, by using techniques from the theory of transport equations with non smooth vector fields, we show that solutions of the incompressible 2D Euler equations obtained from the ones of the 2D incompressible Navier-Stokes equations via the vanishing viscosity limit satisfy a representation formula in terms of the flow of the velocity and that the strong convergence of the vorticity holds. Moreover, we also prove a rate of convergence. The talk is based on results obtained in collaboration with Gianluca Crippa (Univ. Basel) and Gennaro Ciampa (Università Milano Statale).

Resources

• Paper (Website, PDF)

Barbara Nelli (University of L'Aquila)

Title

The Jenkins-Serrin Theorem reloaded.

Abstract

Jenkins and Serrin in the sixties proved a famous theorem about minimal graphs in the Euclidean 3-space with infinite boundary values. After reviewing the classical results, we show how to solve the Jenkins-Serrin problem in a 3-manifold with a Killing vector field. This is a joint work with A. Del Prete and J. M. Manzano.

Resources

• Paper (Website, PDF)

Emanuela Radici (University of L'Aquila)

Title

On curvature and five gradients inequality on manifolds.

Abstract

Introduced almost ten years ago, the five gradients inequality has been used to provide estimates on Sobolev norms of minimizers involving the Wasserstein distance. In conjunction with the JKO scheme, this inequality can grant compactness for the minimizing movement scheme. We investigate the geometric and functional meaning of the five gradients inequality in two generalizations. In the setting of Lie groups the proof naturally suggests that it is a second order optimality condition for the Kantorovich potentials, while in general compact Riemannian manifolds the curvature plays a role. This is a joint work with Simone Di Marino and Simone Murro.

Resources

• Paper (Website, PDF)

Emanuele Tasso (TU Wien)

Title

Rectifiability of a class of integralgeometric measures and applications

Abstract

In his textbook 'Geometric Measure Theory', Federer proposed the following problem: (Q) Is the restriction of the m-dimensional integralgeometric measure to a finite set a m-rectifiable measure? With this motivation, after a brief introduction to the integral geometric measure, I will discuss how rectifiability issues in the spirit of (Q) play an important role in some variational problems. As a sort of unifying theory, I will then introduce a novel class of measures in the euclidean space based upon the idea of slicing. The central result of this talk will follow, which is a sufficient condition for rectifiability in the above class. Two main applications will be shown: the solution to Federer's problem, as well as a novel rectifiability criterion for Radon measures via slicing, the latter being reminiscent of White's rectifiable slices theorem for flat chains. If time permits, I will discuss how to extend the main result to the Riemannian case by means of the notion of transversal family of maps. In the very last part of the talk I will propose some related open problems.

Resources

• Paper (Website, PDF)

Enrico Savi (University of Trento)

Title

Embedded $\mathbb{Q}$-desingularization of real Schubert varieties and application to the relative $\mathbb{Q}$-algebraicity problem

Abstract

In 2020 Parusiński and Rond proved that every algebraic set $X \subset \mathbb{R}^n$ is homeomorphic to an algebraic set $X' \subset \mathbb{R}^n$ which is described globally (and also locally) by polynomial equations whose coefficients are real algebraic numbers. In general, the following problem was widely open: Open Problem. Is every real algebraic set homeomorphic to a real algebraic set defined by polynomial equations with rational coefficients? The aim of my PhD thesis is to provide classes of real algebraic sets that positively answer to above Open Problem. In Chapter 1 I introduce a new theory of real and complex algebraic geometry over subfields recently developed by Fernando and Ghiloni. In particular, the main notion to outline is the so called $\mathbb{R}|\mathbb{Q}$-regularity of points of a $\mathbb{Q}$-algebraic set $X \subset \mathbb{R}^n$. This definition suggests a natural notion of a $\mathbb{Q}$-nonsingular $\mathbb{Q}$-algebraic set $X \subset \mathbb{R}^n$. The study of $\mathbb{Q}$-nonsingular $\mathbb{Q}$-algebraic sets is the main topic of Chapter 2. Then, in Chapter 3 I introduce $\mathbb{Q}$-algebraic approximation techniques a là Akbulut-King developed in collaboration with Ghiloni and the main consequences we proved, that are, versions "over $\mathbb{Q}$" of the classical and the relative Nash-Tognoli theorems. Last results can be found in in Chapters 3 & 4, respectively. In particular, we obtained a positive answer to above Open Problem in the case of compact nonsingular algebraic sets. Then, after extending "over $\mathbb{Q}$" the Akbulut-King blowing down lemma, we are in position to give a complete positive answer to above Open Problem also in the case of compact algebraic sets with isolated singularities in Chapter 4. After algebraic Alexandroff compactification, we obtained a positive answer also in the case of non-compact algebraic sets with isolated singularities. Other related topics are investigated in Chapter 4 such as the existence of $\mathbb{Q}$-nonsingular $\mathbb{Q}$-algebraic models of Nash manifolds over every real closed field and an answer to the $\mathbb{Q}$-algebrization problem for germs of an isolated algebraic singularity. Appendices A & B contain results on Nash approximation and an evenness criterion for the degree of global smoothings of subanalytic sets, respectively.

Resources

• Paper (Website, PDF)